{\textcopyright} 2018 Beaugrand et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Species richness is greater in places where the number of potential niches is high. Consequently, the niche may be fundamental for understanding the arrangement of life and especially, the establishment and maintenance of the well-known Latitudinal Biodiversity Gradient (LBG). However, not all potential niches may be occupied fully in a habitat, as measured by niche vacancy/saturation. Here, we theoretically reconstruct oceanic biodiversity and analyse modeled and observed data together to examine patterns in niche saturation (i.e. the ratio between observed and theoretical biodiversity of a given taxon) for several taxonomic groups. Our results led us to hypothesize that the arrangement of marine life is constrained by the distribution of the maximal number of species{\textquoteright} niches available, which represents a fundamental mathematical limit to the number of species that can co-exist locally. We liken this arrangement to a type of chessboard where each square on the board is a geographic area, itself comprising a distinct number of sub-squares (species{\textquoteright} niches). Each sub-square on the chessboard can accept a unique species of a given ecological guild, whose occurrence is determined by speciation/extinction. Because of the interaction between the thermal niche and changes in temperature, our study shows that the chessboard has more sub-squares at mid-latitudes and we suggest that many clades should exhibit a LBG because their probability of emergence should be higher in the tropics where more niches are available. Our work reveals that each taxonomic group has its own unique chessboard and that global niche saturation increases when organismal complexity decreases. As a result, the mathematical influence of the chessboard is likely to be more prominent for taxonomic groups with low (e.g. plankton) than great (e.g. mammals) biocomplexity. Our study therefore reveals the complex interplay between a fundamental mathematical constraint on biodiversity resulting from the interaction between the species{\textquoteright} ecological niche and fluctuations in the environmental regime (here, temperature), which has a predictable component and a stochastic-like biological influence (diversification rates, origination and clade age) that may alter or blur the former.